[<< wikibooks] Commutative Algebra/Sequences of modules
== Modules in category theory ==

We aim now to prove that if 
  
    
      
        R
      
    
    {\displaystyle R}
   is a ring, 
  
    
      
        R
      
    
    {\displaystyle R}
  -mod is an Abelian category. We do so by verifying that modules have all the properties required for being an Abelian category.
Theorem 10.1:
The category of modules has kernels.
Proof:
For 
  
    
      
        R
      
    
    {\displaystyle R}
  -modules 
  
    
      
        M
        ,
        N
      
    
    {\displaystyle M,N}
   and a morphism 
  
    
      
        f
        :
        M
        →
        N
      
    
    {\displaystyle f:M\to N}
   we define

  
    
      
        ker
        ⁡
        f
        :=
        {
        m
        ∈
        M
        
          |
        
        f
        (
        m
        )
        =
        0
        }
      
    
    {\displaystyle \ker f:=\{m\in M|f(m)=0\}}
  .


== Sequences of augmented modules ==

-category-theoretic comment